# [R] Discrepency between confidence intervals from t.test and computed manually -- why?

David Rossiter rossiter at itc.nl
Wed Jan 4 09:58:51 CET 2006

I am sure there is something simple here I am missing, so please bear
with  me.

It concerns the computation of the confidence interval for a population
mean.

The data are 125 measurements of Cs137 radation, a sample data set from
Davis "Statistics and Data Analysis in Geology" 3rd ed. (CROATRAD.TXT)
------------------
method 1: using textbook definitions: mean \pm se_mean * t-value

mu <- mean(Cs137); n <- length(Cs137)
se.mean <- sqrt(var(Cs137)/n)
# two-tail alphas
alpha <- c(1, 5, 10, 20)/100
# t-values for each tail
t.vals <- qt(1-(alpha/2), n-1)
# name them for the respective alpha
names(t.vals) <- alpha
# low and high ends of the confidence interval
round(ci.low <- mu - se.mean * t.vals, 2)
round(ci.hi <- mu + se.mean * t.vals, 2)

Output:
0.01 0.05  0.1  0.2
5.66 5.81 5.90 5.99

0.01 0.05  0.1  0.2
6.69 6.54 6.46 6.36

-----------------

So for the 95% confidence level I seem to get a CI of 5.81 .. 6.54

------------------
method 2: using t.test.  I am not really testing for any specific mean,
I just want the confidence interval of the mean, which t.test seems to
give to me:

Input:
t.test(Cs137)

Output:

One Sample t-test

data:  Cs137
t = 11.5122, df = 124, p-value < 2.2e-16              <-- not relevant
alternative hypothesis: true mean is not equal to 0   <-- not relevant
95 percent confidence interval:
5.115488 7.239712
sample estimates:
mean of x
6.1776
------------------------------

So with t.test I seem to get a CI of 5.12 .. 7.24 which is considerably
wider than the directly computed interval 5.81 .. 6.54.  Perhaps I am
mis-understanding the CI which t.test is reporting?

Any help would be appreciated.

Thank you.

D G Rossiter
Senior University Lecturer
Department of Earth Systems Analysis (DESA)
International Institute for Geo-Information Science and Earth
Observation (ITC)
Hengelosestraat 99
PO Box 6, 7500 AA Enschede, The Netherlands
mailto:rossiter at itc.nl,  Internet: http://www.itc.nl/personal/rossiter